The effects of the Reynolds number (Re) and thickness on an undulatory self-propelled foil were numerically investigated using the immersed boundary method. Re varied from 50 to 2 × 105, which encompasses the viscous, intermediate, and inertial regimes using a NACA 0012 airfoil. An investigation of the thickness was performed on NACA airfoils with maximum thicknesses of 0.04 ∼ 0.24 at two Re values (5 × 104 and 500). The results indicated that the foil can achieve a higher forward velocity, perform less work, and exhibit a higher propulsive efficiency with increasing Re. However, the effect of Re is asymptotic beyond 5 × 104. Four types of vortex structures exist, and the transition from one regime to another is closely related to hydrodynamic changes. In the thickness study, thinner foils outperformed thicker foils in terms of the forward velocity and input power at both Re values. However, the efficiency related to the conversion of input power into kinetic energy for NACA 0004 was the lowest. An optimum thickness exists that depends on Re. At higher Re, the vortical structure differs for each thickness with the deflection angle, whereas at low Re, the location of the separation point strongly influences the hydrodynamics.

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The drag-thrust transition and wake structures of a pitching foil undergoing combinations of fast and slow pitching are systematically investigated. The foil locomotion having combinations of fast and slow pitching is made by setting a variable s defined as the fraction of the pitching time required on the upper side of the wake centerline. On the other hand, time 1-s is required for the foil to pitch on the lower side of the wake centerline. Compared to the symmetric pitching (s = 0.5) case, the time-mean thrust rapidly increases and the drag-thrust boundary advances with increasing |s − 0.5|. The Kármán vortex street slants and produces thrust when |s − 0.5| is sufficiently large, which supersedes the previous thumb rule that only reverse Kármán vortex street can produce thrust. The faster forward stroke determines the slant direction of the vortex street. The detailed wake structures produced by the pitching foil are discussed, showing how the combined pitching affects vortex dynamics, drag-thrust transition, slant direction, and wake jet. This work provides a physical basis for understanding the hydrodynamics of native swimmers which may be useful to design bio-inspired underwater robots.

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The impact of Strouhal number St (= 0.1–1.0), Reynolds number Re (= 50–2000) and dimensionless wavelength λ (= 0.5–2.0) on the hydrodynamic performance of a travelling wavy foil of a constant length is extensively investigated. The relationship of time-mean thrust with St, Re and λ is presented, suggesting that the propulsive force increases with increasing St, Re and λ. As such, the drag–thrust boundary advances as these parameters increase. A shorter λ makes the thrust steadier while a longer λ enhances the maximum instantaneous thrust. The latter is beneficial for prey to escape from a predator. The fluid added mass caused by the foil oscillation increases with St and λ but declines with Re (<500). Seven types of wake structures produced by the foil are identified, discussed and connected to thrust generation, showing how St, Re and λ affect the fluid dynamics, wake transition, vortex strength, wake jet, velocity, added mass, added damping, power input, efficiency and pressure profiles. The outcome of this work renders a physical basis for understanding the swimming of aquatic animals.